Truncated Fourier Integral

Truncated Fourier Integral

$f\in L^p$ for $1\le p\le 2$. $S_{R}f(x)={\int }_{|\xi |\le R}\hat{f}(\xi )e^{2\pi ix\cdot \xi }d\xi$

$1<p<2$ relies on Riesz-Thorin complex interpolation.

The well-definedness for

Proposition

For $g\in \mathcal{S}$, $S_{R}g\to g$ in $L^p$, with $1\le p<2$, with $\mathcal{S}$ denoting the space of Schwartz Function.